3.166 \(\int \frac {a+b x+c x^2+d x^3}{2+3 x^4} \, dx\)

Optimal. Leaf size=176 \[ -\frac {\left (\sqrt {6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{8\ 6^{3/4}}-\frac {\left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac {\left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}}+\frac {b \tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}}+\frac {1}{12} d \log \left (3 x^4+2\right ) \]

[Out]

1/12*d*ln(3*x^4+2)+1/12*b*arctan(1/2*x^2*6^(1/2))*6^(1/2)-1/48*ln(-6^(3/4)*x+3*x^2+6^(1/2))*(-2*c+a*6^(1/2))*6
^(1/4)+1/48*ln(6^(3/4)*x+3*x^2+6^(1/2))*(-2*c+a*6^(1/2))*6^(1/4)+1/24*arctan(-1+6^(1/4)*x)*(2*c+a*6^(1/2))*6^(
1/4)+1/24*arctan(1+6^(1/4)*x)*(2*c+a*6^(1/2))*6^(1/4)

________________________________________________________________________________________

Rubi [A]  time = 0.14, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1876, 1168, 1162, 617, 204, 1165, 628, 1248, 635, 203, 260} \[ -\frac {\left (\sqrt {6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{8\ 6^{3/4}}-\frac {\left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac {\left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}}+\frac {b \tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}}+\frac {1}{12} d \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2 + d*x^3)/(2 + 3*x^4),x]

[Out]

(b*ArcTan[Sqrt[3/2]*x^2])/(2*Sqrt[6]) - ((Sqrt[6]*a + 2*c)*ArcTan[1 - 6^(1/4)*x])/(4*6^(3/4)) + ((Sqrt[6]*a +
2*c)*ArcTan[1 + 6^(1/4)*x])/(4*6^(3/4)) - ((Sqrt[6]*a - 2*c)*Log[Sqrt[6] - 6^(3/4)*x + 3*x^2])/(8*6^(3/4)) + (
(Sqrt[6]*a - 2*c)*Log[Sqrt[6] + 6^(3/4)*x + 3*x^2])/(8*6^(3/4)) + (d*Log[2 + 3*x^4])/12

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1168

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]

Rule 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 1876

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[(x^ii*(Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii
]*x^(n/2)))/(a + b*x^n), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ
[n/2, 0] && Expon[Pq, x] < n

Rubi steps

\begin {align*} \int \frac {a+b x+c x^2+d x^3}{2+3 x^4} \, dx &=\int \left (\frac {a+c x^2}{2+3 x^4}+\frac {x \left (b+d x^2\right )}{2+3 x^4}\right ) \, dx\\ &=\int \frac {a+c x^2}{2+3 x^4} \, dx+\int \frac {x \left (b+d x^2\right )}{2+3 x^4} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {b+d x}{2+3 x^2} \, dx,x,x^2\right )+\frac {1}{12} \left (\sqrt {6} a-2 c\right ) \int \frac {\sqrt {6}-3 x^2}{2+3 x^4} \, dx+\frac {1}{12} \left (\sqrt {6} a+2 c\right ) \int \frac {\sqrt {6}+3 x^2}{2+3 x^4} \, dx\\ &=\frac {1}{2} b \operatorname {Subst}\left (\int \frac {1}{2+3 x^2} \, dx,x,x^2\right )-\frac {\left (\sqrt {6} a-2 c\right ) \int \frac {\frac {2^{3/4}}{\sqrt [4]{3}}+2 x}{-\sqrt {\frac {2}{3}}-\frac {2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{8\ 6^{3/4}}-\frac {\left (\sqrt {6} a-2 c\right ) \int \frac {\frac {2^{3/4}}{\sqrt [4]{3}}-2 x}{-\sqrt {\frac {2}{3}}+\frac {2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{8\ 6^{3/4}}+\frac {1}{24} \left (\sqrt {6} a+2 c\right ) \int \frac {1}{\sqrt {\frac {2}{3}}-\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac {1}{24} \left (\sqrt {6} a+2 c\right ) \int \frac {1}{\sqrt {\frac {2}{3}}+\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac {1}{2} d \operatorname {Subst}\left (\int \frac {x}{2+3 x^2} \, dx,x,x^2\right )\\ &=\frac {b \tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}}-\frac {\left (\sqrt {6} a-2 c\right ) \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a-2 c\right ) \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}+\frac {1}{12} d \log \left (2+3 x^4\right )+\frac {\left (\sqrt {6} a+2 c\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}-\frac {\left (\sqrt {6} a+2 c\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{4\ 6^{3/4}}\\ &=\frac {b \tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}}-\frac {\left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac {\left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{4\ 6^{3/4}}-\frac {\left (\sqrt {6} a-2 c\right ) \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a-2 c\right ) \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}+\frac {1}{12} d \log \left (2+3 x^4\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.19, size = 164, normalized size = 0.93 \[ \frac {1}{48} \left (-2 \sqrt [4]{6} \tan ^{-1}\left (1-\sqrt [4]{6} x\right ) \left (\sqrt {6} a+2 \left (\sqrt [4]{6} b+c\right )\right )+2 \sqrt [4]{6} \tan ^{-1}\left (\sqrt [4]{6} x+1\right ) \left (\sqrt {6} a-2 \sqrt [4]{6} b+2 c\right )-\sqrt [4]{6} \left (\sqrt {6} a-2 c\right ) \log \left (\sqrt {6} x^2-2 \sqrt [4]{6} x+2\right )+\sqrt [4]{6} \left (\sqrt {6} a-2 c\right ) \log \left (\sqrt {6} x^2+2 \sqrt [4]{6} x+2\right )+4 d \log \left (3 x^4+2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2 + d*x^3)/(2 + 3*x^4),x]

[Out]

(-2*6^(1/4)*(Sqrt[6]*a + 2*(6^(1/4)*b + c))*ArcTan[1 - 6^(1/4)*x] + 2*6^(1/4)*(Sqrt[6]*a - 2*6^(1/4)*b + 2*c)*
ArcTan[1 + 6^(1/4)*x] - 6^(1/4)*(Sqrt[6]*a - 2*c)*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] + 6^(1/4)*(Sqrt[6]*a - 2*
c)*Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2] + 4*d*Log[2 + 3*x^4])/48

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c*x^2+b*x+a)/(3*x^4+2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [A]  time = 0.22, size = 149, normalized size = 0.85 \[ \frac {1}{24} \, {\left (6^{\frac {3}{4}} a - 2 \, \sqrt {6} b + 2 \cdot 6^{\frac {1}{4}} c\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{24} \, {\left (6^{\frac {3}{4}} a + 2 \, \sqrt {6} b + 2 \cdot 6^{\frac {1}{4}} c\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{48} \, {\left (6^{\frac {3}{4}} a - 2 \cdot 6^{\frac {1}{4}} c + 4 \, d\right )} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) - \frac {1}{48} \, {\left (6^{\frac {3}{4}} a - 2 \cdot 6^{\frac {1}{4}} c - 4 \, d\right )} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c*x^2+b*x+a)/(3*x^4+2),x, algorithm="giac")

[Out]

1/24*(6^(3/4)*a - 2*sqrt(6)*b + 2*6^(1/4)*c)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) + 1/2
4*(6^(3/4)*a + 2*sqrt(6)*b + 2*6^(1/4)*c)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) + 1/48*(
6^(3/4)*a - 2*6^(1/4)*c + 4*d)*log(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) - 1/48*(6^(3/4)*a - 2*6^(1/4)*c -
4*d)*log(x^2 - sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3))

________________________________________________________________________________________

maple [A]  time = 0.05, size = 252, normalized size = 1.43 \[ \frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )}{24}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )}{24}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, a \ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}{x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}\right )}{48}+\frac {\sqrt {6}\, b \arctan \left (\frac {\sqrt {6}\, x^{2}}{2}\right )}{12}+\frac {\sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )}{72}+\frac {\sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )}{72}+\frac {\sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, c \ln \left (\frac {x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}{x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}\right )}{144}+\frac {d \ln \left (3 x^{4}+2\right )}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3+c*x^2+b*x+a)/(3*x^4+2),x)

[Out]

1/48*3^(1/2)*6^(1/4)*2^(1/2)*a*ln((x^2+1/3*3^(1/2)*6^(1/4)*2^(1/2)*x+1/3*6^(1/2))/(x^2-1/3*3^(1/2)*6^(1/4)*2^(
1/2)*x+1/3*6^(1/2)))+1/24*3^(1/2)*6^(1/4)*2^(1/2)*a*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x+1)+1/24*3^(1/2)*6^(1/
4)*2^(1/2)*a*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x-1)+1/12*6^(1/2)*b*arctan(1/2*6^(1/2)*x^2)+1/72*3^(1/2)*6^(3/
4)*2^(1/2)*c*arctan(1/6*2^(1/2)*3^(1/2)*6^(3/4)*x+1)+1/72*3^(1/2)*6^(3/4)*2^(1/2)*c*arctan(1/6*2^(1/2)*3^(1/2)
*6^(3/4)*x-1)+1/144*3^(1/2)*6^(3/4)*2^(1/2)*c*ln((x^2-1/3*3^(1/2)*6^(1/4)*2^(1/2)*x+1/3*6^(1/2))/(x^2+1/3*3^(1
/2)*6^(1/4)*2^(1/2)*x+1/3*6^(1/2)))+1/12*d*ln(3*x^4+2)

________________________________________________________________________________________

maxima [A]  time = 2.99, size = 207, normalized size = 1.18 \[ -\frac {1}{144} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} {\left (\sqrt {3} \sqrt {2} c - 2 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} d - 3 \, a\right )} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {1}{144} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} {\left (\sqrt {3} \sqrt {2} c + 2 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} d - 3 \, a\right )} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {1}{72} \, \sqrt {3} {\left (3 \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} a + 2 \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} c - 6 \, \sqrt {2} b\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{72} \, \sqrt {3} {\left (3 \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} a + 2 \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} c + 6 \, \sqrt {2} b\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c*x^2+b*x+a)/(3*x^4+2),x, algorithm="maxima")

[Out]

-1/144*3^(3/4)*2^(3/4)*(sqrt(3)*sqrt(2)*c - 2*3^(1/4)*2^(1/4)*d - 3*a)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + s
qrt(2)) + 1/144*3^(3/4)*2^(3/4)*(sqrt(3)*sqrt(2)*c + 2*3^(1/4)*2^(1/4)*d - 3*a)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3
/4)*x + sqrt(2)) + 1/72*sqrt(3)*(3*3^(1/4)*2^(3/4)*a + 2*3^(3/4)*2^(1/4)*c - 6*sqrt(2)*b)*arctan(1/6*3^(3/4)*2
^(1/4)*(2*sqrt(3)*x + 3^(1/4)*2^(3/4))) + 1/72*sqrt(3)*(3*3^(1/4)*2^(3/4)*a + 2*3^(3/4)*2^(1/4)*c + 6*sqrt(2)*
b)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2^(3/4)))

________________________________________________________________________________________

mupad [B]  time = 5.64, size = 1168, normalized size = 6.64 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2 + d*x^3)/(3*x^4 + 2),x)

[Out]

symsum(log(9*a*b^2 - 864*root(z^4 - (d*z^3)/3 + (a*c*z^2)/24 + (d^2*z^2)/24 + (b^2*z^2)/48 - (a*c*d*z)/144 - (
b^2*d*z)/288 + (b*c^2*z)/288 - (a^2*b*z)/192 - (d^3*z)/432 - (b*c^2*d)/3456 + (a*c*d^2)/3456 + (a^2*b*d)/2304
- (a*b^2*c)/2304 + (b^2*d^2)/6912 + (a^2*c^2)/4608 + d^4/20736 + c^4/13824 + b^4/9216 + a^4/6144, z, k)^2*a -
9*a^2*c - 6*a*d^2 + 9*b^3*x - 6*c^3 + 144*root(z^4 - (d*z^3)/3 + (a*c*z^2)/24 + (d^2*z^2)/24 + (b^2*z^2)/48 -
(a*c*d*z)/144 - (b^2*d*z)/288 + (b*c^2*z)/288 - (a^2*b*z)/192 - (d^3*z)/432 - (b*c^2*d)/3456 + (a*c*d^2)/3456
+ (a^2*b*d)/2304 - (a*b^2*c)/2304 + (b^2*d^2)/6912 + (a^2*c^2)/4608 + d^4/20736 + c^4/13824 + b^4/9216 + a^4/6
144, z, k)*a*d - 144*root(z^4 - (d*z^3)/3 + (a*c*z^2)/24 + (d^2*z^2)/24 + (b^2*z^2)/48 - (a*c*d*z)/144 - (b^2*
d*z)/288 + (b*c^2*z)/288 - (a^2*b*z)/192 - (d^3*z)/432 - (b*c^2*d)/3456 + (a*c*d^2)/3456 + (a^2*b*d)/2304 - (a
*b^2*c)/2304 + (b^2*d^2)/6912 + (a^2*c^2)/4608 + d^4/20736 + c^4/13824 + b^4/9216 + a^4/6144, z, k)*b*c + 12*b
*c*d - 108*root(z^4 - (d*z^3)/3 + (a*c*z^2)/24 + (d^2*z^2)/24 + (b^2*z^2)/48 - (a*c*d*z)/144 - (b^2*d*z)/288 +
(b*c^2*z)/288 - (a^2*b*z)/192 - (d^3*z)/432 - (b*c^2*d)/3456 + (a*c*d^2)/3456 + (a^2*b*d)/2304 - (a*b^2*c)/23
04 + (b^2*d^2)/6912 + (a^2*c^2)/4608 + d^4/20736 + c^4/13824 + b^4/9216 + a^4/6144, z, k)*a^2*x + 864*root(z^4
- (d*z^3)/3 + (a*c*z^2)/24 + (d^2*z^2)/24 + (b^2*z^2)/48 - (a*c*d*z)/144 - (b^2*d*z)/288 + (b*c^2*z)/288 - (a
^2*b*z)/192 - (d^3*z)/432 - (b*c^2*d)/3456 + (a*c*d^2)/3456 + (a^2*b*d)/2304 - (a*b^2*c)/2304 + (b^2*d^2)/6912
+ (a^2*c^2)/4608 + d^4/20736 + c^4/13824 + b^4/9216 + a^4/6144, z, k)^2*b*x + 72*root(z^4 - (d*z^3)/3 + (a*c*
z^2)/24 + (d^2*z^2)/24 + (b^2*z^2)/48 - (a*c*d*z)/144 - (b^2*d*z)/288 + (b*c^2*z)/288 - (a^2*b*z)/192 - (d^3*z
)/432 - (b*c^2*d)/3456 + (a*c*d^2)/3456 + (a^2*b*d)/2304 - (a*b^2*c)/2304 + (b^2*d^2)/6912 + (a^2*c^2)/4608 +
d^4/20736 + c^4/13824 + b^4/9216 + a^4/6144, z, k)*c^2*x + 9*a^2*d*x + 6*b*d^2*x - 6*c^2*d*x - 144*root(z^4 -
(d*z^3)/3 + (a*c*z^2)/24 + (d^2*z^2)/24 + (b^2*z^2)/48 - (a*c*d*z)/144 - (b^2*d*z)/288 + (b*c^2*z)/288 - (a^2*
b*z)/192 - (d^3*z)/432 - (b*c^2*d)/3456 + (a*c*d^2)/3456 + (a^2*b*d)/2304 - (a*b^2*c)/2304 + (b^2*d^2)/6912 +
(a^2*c^2)/4608 + d^4/20736 + c^4/13824 + b^4/9216 + a^4/6144, z, k)*b*d*x - 18*a*b*c*x)*root(z^4 - (d*z^3)/3 +
(a*c*z^2)/24 + (d^2*z^2)/24 + (b^2*z^2)/48 - (a*c*d*z)/144 - (b^2*d*z)/288 + (b*c^2*z)/288 - (a^2*b*z)/192 -
(d^3*z)/432 - (b*c^2*d)/3456 + (a*c*d^2)/3456 + (a^2*b*d)/2304 - (a*b^2*c)/2304 + (b^2*d^2)/6912 + (a^2*c^2)/4
608 + d^4/20736 + c^4/13824 + b^4/9216 + a^4/6144, z, k), k, 1, 4)

________________________________________________________________________________________

sympy [B]  time = 13.07, size = 580, normalized size = 3.30 \[ \operatorname {RootSum} {\left (165888 t^{4} - 55296 t^{3} d + t^{2} \left (6912 a c + 3456 b^{2} + 6912 d^{2}\right ) + t \left (- 864 a^{2} b - 1152 a c d - 576 b^{2} d + 576 b c^{2} - 384 d^{3}\right ) + 27 a^{4} + 72 a^{2} b d + 36 a^{2} c^{2} - 72 a b^{2} c + 48 a c d^{2} + 18 b^{4} + 24 b^{2} d^{2} - 48 b c^{2} d + 12 c^{4} + 8 d^{4}, \left (t \mapsto t \log {\left (x + \frac {- 41472 t^{3} a^{2} c + 82944 t^{3} a b^{2} + 27648 t^{3} c^{3} + 5184 t^{2} a^{3} b + 10368 t^{2} a^{2} c d - 20736 t^{2} a b^{2} d + 10368 t^{2} a b c^{2} - 6912 t^{2} b^{3} c - 6912 t^{2} c^{3} d + 648 t a^{5} - 864 t a^{3} b d - 1728 t a^{3} c^{2} + 3888 t a^{2} b^{2} c - 864 t a^{2} c d^{2} + 864 t a b^{4} + 1728 t a b^{2} d^{2} - 1728 t a b c^{2} d + 864 t a c^{4} + 1152 t b^{3} c d + 864 t b^{2} c^{3} + 576 t c^{3} d^{2} - 54 a^{5} d + 270 a^{4} b c - 270 a^{3} b^{3} + 36 a^{3} b d^{2} + 144 a^{3} c^{2} d - 324 a^{2} b^{2} c d + 24 a^{2} c d^{3} - 72 a b^{4} d + 180 a b^{3} c^{2} - 48 a b^{2} d^{3} + 72 a b c^{2} d^{2} - 72 a c^{4} d - 72 b^{5} c - 48 b^{3} c d^{2} - 72 b^{2} c^{3} d + 72 b c^{5} - 16 c^{3} d^{3}}{81 a^{6} - 54 a^{4} c^{2} + 432 a^{3} b^{2} c - 216 a^{2} b^{4} - 36 a^{2} c^{4} + 288 a b^{2} c^{3} - 144 b^{4} c^{2} + 24 c^{6}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**3+c*x**2+b*x+a)/(3*x**4+2),x)

[Out]

RootSum(165888*_t**4 - 55296*_t**3*d + _t**2*(6912*a*c + 3456*b**2 + 6912*d**2) + _t*(-864*a**2*b - 1152*a*c*d
- 576*b**2*d + 576*b*c**2 - 384*d**3) + 27*a**4 + 72*a**2*b*d + 36*a**2*c**2 - 72*a*b**2*c + 48*a*c*d**2 + 18
*b**4 + 24*b**2*d**2 - 48*b*c**2*d + 12*c**4 + 8*d**4, Lambda(_t, _t*log(x + (-41472*_t**3*a**2*c + 82944*_t**
3*a*b**2 + 27648*_t**3*c**3 + 5184*_t**2*a**3*b + 10368*_t**2*a**2*c*d - 20736*_t**2*a*b**2*d + 10368*_t**2*a*
b*c**2 - 6912*_t**2*b**3*c - 6912*_t**2*c**3*d + 648*_t*a**5 - 864*_t*a**3*b*d - 1728*_t*a**3*c**2 + 3888*_t*a
**2*b**2*c - 864*_t*a**2*c*d**2 + 864*_t*a*b**4 + 1728*_t*a*b**2*d**2 - 1728*_t*a*b*c**2*d + 864*_t*a*c**4 + 1
152*_t*b**3*c*d + 864*_t*b**2*c**3 + 576*_t*c**3*d**2 - 54*a**5*d + 270*a**4*b*c - 270*a**3*b**3 + 36*a**3*b*d
**2 + 144*a**3*c**2*d - 324*a**2*b**2*c*d + 24*a**2*c*d**3 - 72*a*b**4*d + 180*a*b**3*c**2 - 48*a*b**2*d**3 +
72*a*b*c**2*d**2 - 72*a*c**4*d - 72*b**5*c - 48*b**3*c*d**2 - 72*b**2*c**3*d + 72*b*c**5 - 16*c**3*d**3)/(81*a
**6 - 54*a**4*c**2 + 432*a**3*b**2*c - 216*a**2*b**4 - 36*a**2*c**4 + 288*a*b**2*c**3 - 144*b**4*c**2 + 24*c**
6))))

________________________________________________________________________________________